Application To Ballistic Missiles: A ballistic flight is that in which a projectile is given an initial push and is then allowed to move freely due to inertia and under the action of gravity. An un-powered and un-guided missile is called a ballistic missile and the path followed by it is called ballistic trajectory.

As discussed before, a ballistic missile moves in a way that is the result of the superposition of two independent motion: a straight line inertial flight in the direction of the launch and a vertical gravity fall. By law of inertia, an object should sail straight off in the direction thrown, at constant speed equal to its initial speed particularly in empty space. But the downward force of gravity will alter straight path into a curved trajectory. For short ranges and flat Earth approximation, the trajectory is parabolic but the dragless ballistic trajectory for spherical Earth should actually be elliptical. At high speed and for long trajectories the air friction is not negligible and some times the force of air friction is more than gravity. It affects both horizontal as well as vertical motions. Therefore, it is completely unrealistic to neglect the aerodynamic forces.

The shooting of a missile on a selected distant spot is a major element of warfare. It undergoes complicated motions due to air friction and wind etc. consequently the angle of projection can not be found by the geometry of the situation at the moment of launching. The actual flights of missiles are worked out to high degrees of precision and the result were contained in tabular form. The modified equation of trajectory is too complicated to be discussed here. The ballistic missiles are useful only for short ranges. For long ranges and greater precision, powered and remote control guided missiles are used.

 Do You Know? For an angle less than 45⁰, the height reached by the projectile and the range both will be less. When the angle of projectile is larger than 45⁰, the height attained will be more but the range is again less.

Example 3.7: A ball is thrown with a speed of 30 ms-1 in a direction 30⁰ above the horizon. Determine the height to which it rises, the time of flight and the horizontal range.

Solution: initially

Example 3.8: In example 3.7 calculate the maximum range and the height reached by the ball if the angles of projection are (i) 45⁰ (ii) 60⁰.

Solution:

(i)                  Using the equation for height and range we have

(ii)                Using the equation for height and range we have

 For your information In the presence of air friction the trajectory of a high speed projectile fall short of a parabolic path.